Problem: All of the 5th grade teachers and students from Springer went on a field trip to an archaeology museum. Tickets were $$5.50$ each for teachers and $$3.00$ each for students, and the group paid $$46.50$ in total. A few weeks later, the same group visited a natural history museum where the tickets cost $$11.00$ each for teachers and $$12.50$ each for students, and the group paid $$158.00$ in total. Find the number of teachers and students on the field trips.
Answer: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${5.5x+3y = 46.5}$ ${11x+12.5y = 158}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-2$ ${-11x-6y = -93}$ ${11x+12.5y = 158}$ Add the top and bottom equations together. $ 6.5y = 65 $ $ y = \dfrac{65}{6.5}$ ${y = 10}$ Now that you know ${y = 10}$ , plug it back into $ {5.5x+3y = 46.5}$ to find $x$ ${5.5x + 3}{(10)}{= 46.5}$ $5.5x+30 = 46.5$ $5.5x = 16.5$ $x = \dfrac{16.5}{5.5}$ ${x = 3}$ You can also plug ${y = 10}$ into $ {11x+12.5y = 158}$ and get the same answer for $x$ ${11x + 12.5}{(10)}{= 158}$ ${x = 3}$ There were $3$ teachers and $10$ students on the field trips.